>The litmus test will be when, if ever, someone wins a Fields using a proof-assistant in an essential way.
You're assuming that the point of interactive theorem provers is to discover new mathematics. While that's an interesting research area, it seems like the more practical application is verifying proofs one has already discovered through other means.
Exactly this. LLMs really aren't built for discovering new mathematics, especially _interesting_ new mathematics. They're built to try the most obvious patterns. When that works, it's pretty much by definition not interesting.
What LLMs are good at is organizing concepts, filling in detail, and remembering to check corner cases. So their use should help mathematicians to get a better handle on what's terra firma and what's still exploration. Which is great. Proof by it-convinced-other-mathematicians doesn't have a flawless track record. Sometimes major theorems turn out to be wrong or wrong-as-stated. Sometimes they're right, but there's never been a complete or completely correct proof in the literature. The latter case is actually quite common, and formal proof is just what's needed.
LLMs and interactive theorem provers are vastly different. There are AI models that come up with workable formal proofs for ITPs but these aren't your usual frontier models, they're specifically trained for this task.
ITPs are far older than LLMs in general, sure, but that's a pedantic distraction. What everyone is talking about here (both the comments, and the article) are ITPs enriched with LLMs to make the "smart" proof assistants. The LLMs used in ITPs are not vastly different from the usual chatbots and coding assistants. Just a different reinforcement learning problem, no fundamental change in their architecture.
Of course, once LLMs are really good at that, they can be set loose on the entire historical math literature, all 3.5M papers worth. And then LLMs can be trained on these formalized results (the ones that turn out upon attempted formalization to have been correct.)
How good do you think AI will be at proving new results given that training set?
Math is going to change, and change massively. There's a lot of whistling past the graveyard going on from those who are frightened by this prospect.
Relevant philosophy paper: "The Vulnerable World Hypothesis" by Nick Bostrom [0].
In that paper, Bostrom floats the idea that it might be in humanity's best interest to have a strong global government with mass surveillance to prevent technological catastrophes. It's more of a thought experiment than a "we should definitely do this" kind of argument, but it's worth taking the idea seriously and thinking hard about what alternatives we have for maintaining global stability.
Cheap hypersonics don't threaten global stability, they threaten global hegemony. Which is really what I suspect irks most people afraid of them.
We've seen a shift towards cheap offensive capacity that gives middle powers or even smaller actors the capacity to hit hegemons where it hurts, very visible in Ukraine and the Middle East now. This leads to instability only temporarily until you end up in a new equilibrium where smaller players will have significantly more say and capacity to retaliate, effectively a MAD strategy on a budget for everyone.
GP's point was broader than that, it was about technological progress and the possibility of terrorist groups or mentally ill individuals getting their hands on weapons that can easily kill millions of people. That's also what the paper I linked is about.
Consider a future where individuals can relatively easily engineer a pathogen or manufacture a nuclear weapon. It's not hard to imagine how that would threaten global stability.
It has been proven that recurrent neural networks are Turing complete [0]. So for every computable function, there is a neural network that computes it. That doesn't say anything about size or efficiency, but in principle this allows neural networks to simulate a wide range of intelligent and creative behavior, including the kind of extrapolation you're talking about.
I think you cannot take the step from any turing machine being representable as a neural network to say anything about the prowess of learned neural networks instead of specifically crafted ones.
I think a good example are calculations or counting letters: it's trivial to write turing machines doing that correctly, so you could create neural networks, that do just that. From LLM we know that they are bad at those tasks.
So for every computable function, there is a neural network that computes it. That doesn't say anything about size or efficiency
It also doesn't say anything about finding the desired function, rather than a different function which approximates it closely on some compact set but diverges from it outside that set. That's the trouble with extrapolation: you don't know how to compute the function you're looking for because you don't know anything about its behaviour outside of your sample.
No, but unless you find evidence to suggest we exceed the Turing computable, Turing completeness is sufficient to show that such systems are not precluded from creativity or intelligence.
I believe that quantum oracles are more powerful than Turing oracles, because quantum oracles can be constructed, from what I understand, and Turing oracles need infinite tape.
Our brains use quantum computation within each neuron [1].
The difference is quantum oracles can be constructed [1] and Turing oracle can't be [2]: "An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an entity unspecified by Turing "apart from saying that it cannot be a machine" (Turing (1939)."
This is meaningless. A Turing machine is defined in terms of state transitions. Between those state transitions, there is a pause in computation at any point where the operations takes time. Those pauses are just not part of the definition because they are irrelevant to the computational outcome.
And given we have no evidence that quantum oracles exceeds the Turing computable, all the evidence we have suggests that they are Turing machines.
Turing machines grew from the constructive mathematics [1], where proofs are constructions of the objects or, in other words, algorithms to compute them.
Saying that there is no difference between things that can be constructed (quantum oracles) and things that are given and cannot be constructed (Turing oracles - they are not even machines of any sort) is a direct refutation of the very base of the Turing machine theoretical base.
That's an irrelevant strawman. It tells us nothing about how create such a system ... how to pluck it out of the infinity of TMs. It's like saying that bridges are necessarily built from atoms and adhere to the laws of physics--that's of no help to engineers trying to build a bridge.
And there's also the other side of the GP's point--Turing completeness not necessary for creativity--not by a long shot. (In fact, humans are not Turing complete.)
No, twisting ot to be about how to create such a system is the strawman.
> Turing completeness not necessary for creativity--not by a long shot.
This is by far a more extreme claim than the others in this thread. A system that is not even Turing complete is extremely limited. It's near impossible to construct a system with the ability to loop and branch that isn't Turing complete, for example.
>(In fact, humans are not Turing complete.)
Humans are at least trivially Turing complete - to be Turing complete, all we need to be able to do is to read and write a tape or simulation of one, and use a lookup table with 6 entries (for the proven minimal (2,3) Turing machine) to choose which steps to follow.
Maybe you mean to suggest we exceed it. There is no evidence we can.
> P.S. everything in the response is wrong ... this person has no idea what it means to be Turing complete.
I know very well what it means to be Turing complete. All the evidence so far, on the other hand suggests you don't.
> An infinite tape. And to be Turing complete we must "simulate" that tape--the tape head is not Turing complete, the whole UTM is.
An IO port is logically equivalent to infinite tape.
> PDAs are not "extremely limited", and we are more limited than PDAs because of our very finite nature.
You can trivially execute every step in a Turing machine, hence you are Turing equivalent. It is clear you do not understand the subject at even a basic level.
> You can trivially execute every step in a Turing machine, hence you are Turing equivalent. It is clear you do not understand the subject at even a basic level.
LOL. Such projection. Humans are provably not Turing Complete because they are guaranteed to halt.
Judging from what I read, their work is subject to regular hardware constraints, such as limited stack size. Because paper describes a mapping from regular hardware circuits to the continuous circuits.
As an example, I would like to ask how to parse balanced brackets grammar (S ::= B <EOS>; B ::= | BB | (B) | [B] | {B};) with that Turing complete recurrent network and how it will deal with precision loss for relatively short inputs.
Paper also does not address training (i.e., automatic search of the processors' equations given inputs and outputs).
Is there actually a use case for graphing calculators anymore? Desmos provides a great graphing program for free in a web browser. In any professional capacity you would be using MATLAB, Mathematica, or the scientific Python ecosystem.
I mostly remember playing games on my TI-84 in high school. We used it in class maybe once or twice. None of my college classes allowed graphing calculators on tests, so ironically I had to buy a "dumb" calculator even though I owned the fancy one.
I don't think there was ever a solid use case for graphing calculators in school, at least not in my experience? The curriculum didn't make good use of them and I'm not convinced it could have. There's little value in having every kid in the classroom replicate the same plot of y = sin(x) or whatever on a tiny screen. And other than such demonstrations... what are you gonna do with it? It was never flexible or powerful enough for serious math. You weren't going to run circuit or physics simulations on a TI-89.
There are other features that can be useful - scientific notation, symbolic solver, unit conversions, etc - but graphing as such always seemed like a gimmick.
I think it's more of a not-entirely-rational appeal to parents: "if my kid has a top-notch calculator for high school / college, maybe they're gonna be better at math". And kids did not object, but in the end, mostly just sideloaded games and horsed around.
> You weren't going to run circuit or physics simulations on a TI-89.
Well, I wrote a couple of programs that were useful for quite a while. They involved electromagnetism and changing frames of reference. I definitely was able to do quite a lot of Physics with my Ti-89.
>DR is a Danish public-service radio and television broadcasting company. Founded in 1925 as a public-service organization, it is Denmark's oldest and largest electronic media enterprise.
Humans are complex. It's possible for someone to want to do good and at the same time want to promote/market their product and make a profit. I don't see a contradiction there.
How do you call a marketing campaign that does not deliver on what it promised? I have no problem with anthropic trying to create good will around their products but this particular campain aiming to find good will around people doing open source was an outright lie that did not deliver what it promised and this was all done on HN.
When a company lies for something that trivial, it does not inspire trust
It's an outright lie because they haven't greenlit your personal project after two weeks? Did it occur to you that maybe they just got a lot of applications and are prioritizing other projects or still working through a backlog?
This is a completely nostalgic, one-sided view of the urban vs. rural divide. It's also ignorant of the data on the impact of cities on nature.
Cities have a lower carbon footprint per capita, lower land use per capita, people own fewer cars, use public transportation more often. If everyone lived in a city, nature would be better off.
People live in cities because they are vibrant, they have culture, the arts, intellectuals, innovation, etc. Yes there are areas with high traffic and noise, but there are also quiet neighborhoods where everyone walks everywhere, you can pop into a bar or a cafe on every corner, eat 20 different types of cuisine, go to a book store, go see a show on any night of the week.
Your picture of the friendly villagers might be true in your experience, but in reality a lot of those people are nasty when they encounter any kind of cultural diversity or difference.
Yeah, but unfortunately false nostalgia for subsistence farming is widespread and has traction in the discourse. I guess it's probably because every American who ever suffered from that lifestyle is dead, in other words the same reason that it is now increasingly popular to die from measles.
In reality, the laser-leveled, fully-automated, county-scale factory farm is the only reason anyone on this forum has ever experienced the phenomenon known as "free time".
Farming is harder than people who haven't done it think, and surviving on the production of only your family's property is really, really hard. Source: I grew up in very rural areas, and I've seen what it entails. My grandparents lived through the Great Depression in a farming community on a homestead.
However, I don't think that's the ask, here. You don't have to choose MEGASUPERTROPOLIS or remote solo farm. There's a huge gradient between the two.
It's possible to put a little effort into gardening, share with your community, and massively reduce the overall cost of food while still having free time.
> However, I don't think that's the ask, here. You don't have to choose MEGASUPERTROPOLIS or remote solo farm. There's a huge gradient between the two.
The gradient is where you start to consume a lot of carbon unnecessarily.
It’s an extremely privileged kind of nostalgia. Only the wealthy can romanticize poverty.
The solution to nostalgia for farming is for people to try it.
I used to live in Asheville NC years ago. There were a lot of hippie back to the land types that came there to find themselves. It’s that kind of place. Sometimes they’d decide to become farmers. This usually lasted at best one year. Sometimes it lasted weeks.
There's a book about being the child of back-to-the-land hippies called "Against the Country" that I enjoyed and recommend to all who might be tempted.
Some of the groups around Asheville were "eco villages" that could get rather culty, including the kinds of high control abuses that occur in cults. All groups weren't like this but I recall hearing some crazy stories involving shaming people into giving money to the group (while the higher ups are "more equal" than everyone else) and sexual exploitation.
There was one case where the group leader/organizer ran away with a bunch of cash, and one or two people tried to sue but they'd set up some kind of shell company structure and had conned people into signing things that made it hard to sue or press criminal charges. I think they also left the other members on the hook for a property lease, since the things they'd been conned into signing did that too. It looked a lot like an explicit long con targeting the hippie trust fund kids that would come around. Sex abuse too apparently.
I heard more than one story that mentioned members of the Hells Angels, at least allegedly.
This is another aspect of small insular communities in the good old days that the trads and anarcho-primitivists (left-trads) don't talk about. It still happens in small communities today, but a lot less due to the presence of larger scale nation states with laws and law enforcement. If the sheriff in your tiny rural town rapes you, you could try to bring charges at the state level or at least sue. Worst case you can move, and with a global money system you can sell your home and take the cash with you. Without that larger context, there is no recourse. If your local tribal chief, town sheriff, or cult leader wants to take your stuff, pressure you into polygamy, or rape your kids, there's nothing you can do about it. Back to nature!
Albeit I feel like OP was right on something else: his grandparents weren't heavy consumers, but that transcends city vs rural debates.
I see us modern people, except very old folks being extremely heavy consumers.
Sometime I pay attention to my friends and relatives and how much do they consume.
E.g. I spent 5 days with my mother in December at my grandmas and I've noticed that she just bought stuff non-stop, but her metric is money, not "stuff".
So, e.g., she bought a new pillow for my grandma even though my grandma didn't need/want one (she doesn't use it), bought plenty of plastic toys for her own dog, bought a set of new dishes just because the old ones were old, changed her worn phone leather case for a new one, bought plenty of Christmas lights because she didn't want to dig for the old ones she couldn't quickly find, bought some kind of table hook for purses for herself and her friends, etc, etc.
At the end of the week she didn't even spend 250 euros (her metric), so she doesn't realizes, yet, the amount of borderline useless stuff she bought was major and her ecological impact quite huge, especially for how little to none the payoff or utility is.
I had colleagues in my office, back when I was in the office, that just had Amazon packages coming every single day...And here's a smartphone holder, here's some gadget that keeps your mouse cord, here's a yet a new pedal for the drums, here's a set of pens, here's a rubber duck to talk to when debugging, etc, etc.
I mean, I have even a difficult time pointing out that there's something wrong with any of those items per se in isolation, but when it's a lifestyle of non-stop consistent consumerism I think the trend is worrying.
There's so many things that are so cheap nowadays that it's hard to say "why not?", yet they feed into this endless life style that's toxic for the planet but feeds this neverending more more and more.
> Your picture of the friendly villagers might be true in your experience, but in reality a lot of those people are nasty when they encounter any kind of cultural diversity or difference.
Cities are all the things you describe, including the myopic perspective and the inflexibility on display in this quote.
And thats why city states - which is what effectively those you describe are - eventually collapse into "dark ages" and future generation of people can't understand why
That's only because cities outsource their carbon intensive activities. There is no "divide" here. It's one planet and focusing on the wrong categories has destroyed your ability to reason.
That's simply not true. The per capita emissions account for things that are produced outside of cities like food. The primary sources of emissions from individual human activities are food, energy production, and transportation. High density areas are more efficient at providing people's heating and transportation needs. As far as I know, people in cities don't eat more food than people in other areas, if you control for income/standard of living.
It's not only because cities outsource carbon intensive activities. Sure, there's some of that with farming, mining, etc. that must be done elsewhere. But there's also lots of savings from things like residents walking/biking/using public transit instead of driving, living in more efficient apartments, etc. The suburbs are pretty wasteful, they don't generate anything unique and they just waste more resources.
But you do realize that all the positives are mostly hedonistic ?
Yeah, there are more places to enjoy yourself and have fun, more entertainment.
I'm passionate about going out to clubs, electronic music events, concerts, restaurants, flying around on a plane or driving my car on the endless roads.
All of this is great, but according to TFA and my own experience, we're absolutely shitting on the natural world to have our nice drink or exotic food which will be gone from our system in 12h.
We've 'borrowed' from the future generations to have our fun and I'm not sure it's all worth the price.
Don't project the emptiness of your existence on the rest of us. Cities are the only reason we have orchestras and ballets, vibrant sports leagues, and other things central to family life. I do not want my kids to have to live in a "village" too small to field a brass quintet.
As someone who is sensitive to noise, rural areas bother me far more than urban ones. Traffic is low-frequency and only rarely annoying, but the few times I have lived in rural areas or gone camping, I have been woken up repeatedly by the horrific screeching of birds. Louder, shriller, less predictable than any city noise.
There are birds in cities too, and they are annoying, but they are thankfully drowned out by the cars.
> Cities have a lower carbon footprint per capita, lower land use per capita, people own fewer cars, use public transportation more often. If everyone lived in a city, nature would be better off.
I think that's apples to oranges. If we didn't have cities, we also wouldn't have eight billion people in the world.
A better question for the parent is how do you enforce that vision of everyone living on their 20 acres in harmony with nature? This is not something that capitalism or some other -ism does to us. Your neighbor will have children, these children will have children, and before long, you have a settlement of 50 people on these 20 acres, most certainly no longer living in harmony with nature. At that point, they must build infrastructure. That infrastructure may be feasible to build if they pool their resources with the neighbors. Boom, you have a village, then a town, then a city.
So what's the solution here? Do we forcibly sterilize people? Lock them up if they have children? What's the anti-growth strategy we're actually advocating for?
Because the habitable surface of the planet is less than 100 million square kilometers and only a fraction of that is suitable for subsistence farming. The only reason we can accommodate 8 billion is that the majority of them live in high-density settlements and that food is grown on an industrial scale elsewhere.
This is obviously not a reversible trend. People having close proximity to one another, creating economies of scale where everyone does what they are best at instead of everyone doing everything for themselves is what allows big cities to be possible.
I'm sure all of this was inevitable as there likely hasn't ever been a time where humans were not getting together to form communities when it was beneficial to do so.
So what's the solution here? Do we forcibly sterilize people? Lock them up if they have children? What's the anti-growth strategy we're actually advocating for?
You don't need to. Fertility rate per woman in wealthy countries decreases every year.
> Do we forcibly sterilize people? Lock them up if they have children?
Oh give me a break. Developed countries have had below-replacement fertility rates for decades.
If your goal is to reduce birth rates, we do the things that we already know do that naturally: comprehensive, fact-based sex ed; cheap and easy access to contraceptives; social safety nets and support; etc etc etc.
It's not a mystery and it's not even difficult. You don't need to jump to straight to abhorrent crimes against humanity.
Another datapoint: I'm a 20+ year off-and-on user of Linux who was never able to switch to desktop Linux for daily use because of the graphics issues. After recently deciding to abandon Windows for good and suffer whatever problems I would have with Linux, I was pleasantly surprised by the experience on Fedora/Wayland/Gnome. No issues with high DPI, per-monitor fractional scaling, tearing, bad performance, etc. which have plagued my Linux experiences in the past. There are still minor issues with Nvidia drivers, but this is very likely the fault of Nvidia and not the OSS community.
Whatever ideological debates there are underneath the X vs. Wayland divide, ultimately what I care about is things working as well as or better than other mainstream operating systems, and Wayland seems to deliver on that.
Teaching and research should be decoupled. Professors are hired and granted tenure primarily based on their ability to produce original research. The skillsets are different; often good researchers are bad teachers, and good teachers are bad researchers.
There is a case to be made that teaching improves the understanding and insight of the teacher which in turn can increase their research ability. For starters, it provides a less boring way of drilling fundamentals. But more importantly, having to answer questions from students which very likely will be coming from odd and unexpected directions, helps the teacher clarify their thinking. It could well be that one of these odd questions, the answer for which the teacher takes for granted, may actually hold some insight or raise questions into what they are working on outside of class.
In a similar vein, it is recommended that if you are in a business meeting you hear what the junior positions have to say about something first and work your way up the chain of command rather than the other way around due to the junior positions being less familiar with internal processes and thus more likely to flag or suggest something completely out of left field that the higher ups might miss.
I tend to agree that teaching can clarify one's ideas, but I don't think the benefits are equal across the board. I think the argument for benefits to research are stronger when it comes to supervising graduate students and teaching seminars. I'm far less convinced that we should have math professors teaching Calc 1 if they're not really passionate about it, and I'm especially not in favor of tying up their salary and performance evaluation with it.
Note, I'm saying all of this as someone outside of academia who is passionate about science and had a very mixed bag of teachers in undergrad.
This used to be the case: research was conducted mostly at academic institutions that did not provide degrees [1]. The "research university" is a relatively new thing
Interesting read. I always wondered from where did the idea about "thesis" & other "extra-circular" activities come from, for both students and professors.
Nowadays, promotions of professors for different levels (Assistant, Associate, Professor) is solely dependent on number of papers they are publishing in Q1 journals.
But the research maybe entirely bogus, same ideas repurposed hundreds of times by different professors.
The entire concept about "systematic knowledge" has gone downhill.
Even more important than the papers is whether you can raise the money required to fund your lab which produces your prestigious journal papers. And the further you go down the league table the less important the "prestigious" part gets.
Absolutely not. You could argue this for entry level lectures, but not at the PhD level. PhD is learning how to do original research, how could you separate teaching that from doing that?
That's not entirely the case in Germany. Applicants need to give a lecture which is public. Usually members of the student union will be present and will have a say later within the hiring committee about the quality of teaching.
But I do agree that the ability to produce and procure research is not at all coupled with the ability to teach.
This is like saying peasants growing vegetables in the field should not mix with philosophers questioning the secrets of the Universe.
Problem is most research is just pissing in the wind. No real results. Show me the cure for cancer. Show me the warp engine.
So it's very nice to sit in their ivory tower doing ivory tower stuff while the peasants feed them with the vegetables they grow plowing the fields.
In reality, let them also teach. That's real, palpable work. I can't do all nice things and never touch shit work, so should professors because unless they cure cancer or invent the warp engine now, they are not a privileged cast.
A lot of academics like teaching and think they benefit from it though. Richard Feynman thought so, and many academics I have met seem to as well. Not all.
I think in some subjects (e.g. literature) the greater prestige of research leads to a lot of pointless research and we need more teaching.
Of course, there are many good researches who are bad teachers. I am not so sure about vice-versa, but, nonetheless good teaching should be rewarded more, as should the ability to communicate knowledge in other ways (e.g. by writing books).
I would suggest 'more loosely coupled' perhaps, which in many cases they already are. Students may not want a surly, disinterested professor forced upon them because of a teaching requirement, but they won't want to miss out on great teachers forced to choose between teaching and researching. And as expressed by others, teaching and research reinforce each other to some degree. Teaching tests conceptual understanding and research incentivizes maintaining an understanding that tracks the knowledge limits of the field.
But this exists in many educational + research institutions already. Where it runs into problems is in resource constrained environments, where there aren't the budgets to support research-only positions that aren't 'less than' roles or the institution can't support the grant ambitions of highest 'performing' teaching researchers, stuck that they don't go to those institutions and more less research-focused (or at least smaller grant value) teachers populate those schools.
I'm trying not to make any value-judgements here, so please ignore and bias vibes that gives off.
I've worked with good and bad at both. Some of the most difficult problems when you have students who have had excellent teachers and then get dropped into the real world. If they don't learn themselves how to apply what they're learning (the other side of the coin of training) then they're often no better than an llm stuck in a loop, they know the textbook but don't know the gray areas...
Also professors and researchers are required to be able to communicate otherwise they're useless to the field. They need to better.
I'm not saying every lecturer will hold any interest in every lecture course. I've had the ones who are there lecturing core material to avoid the dept losing its accreditation and I've done electives where the professor is off the wall and spends half of the time going on about their research instead of the address material (fun but painful come exam time).
In this sort of case the “teaching” happening with graduate assistants is teaching how to do research. That’s inextricably part of the job of a research professor, is to teach others how to do the job.
Imagine a future classroom defined by elaborate plays performed by curious parents, all on advanced adjacent learning paths themselves. An intertwined learning structure that just keeps going up. At higher levels, instead of having the researcher with their head in the books communicating, they’ll have a whole team of people translating their knowledge into a production fit for antiquity - directors, diverse range of talents, charismatic performers, etc.
Assuming we have time to do this in some post-having-jobs world, of course.
I agree. I studied at one of Europe's top "research universities". The most important courses often had the most famous profs, who also were either bad teachers or put no efford into their teaching.
On the other hand the course I learnt the most from was taught by a passionate lecturer who isn't even a professor, and he gets the highest ratings every year. (Yes we have a "rate my prof" event)
So, I'm no professor, but as an ten year post-doc (unfortunately) I can say that most university groups benefit from both types. Again, the problem fundamentally is funding and the wrong incentives, as it always has been from before I entered grad school till now.
"Teaching" in OPs context probably doesn't mean lecturing, but 1:1 sessions with Junior Researchers from Master's Thesis upwards to PhD Candidates and Postdocs.
You probably need to step outside of your US-centric bubble if you are to comment on how university works outside of the US. There was a fairly large clue in the parent comment.
"Often good researchers are bad teachers, and good teachers are bad researchers" is a statement about humans, not a specific country, as far as I can tell. Sure, I happened to use the word "tenure" which is generally used in a U.S. context but you should be able to take a charitable reading of what I said and understand the broader point.
To my knowledge the view is correct for places outside the US.
UK universities do currently hire people to do research and teach. And tenure is based on research not teaching. Teaching is seen as something that funds the operation to an extent. Some are excellent teachers. Some merely provide the material.
It works as is because researchers are not meaningfully impacted by having to do a few hours a week. And student get access to people in touch with the field. But it is not optimal having people who often are not good at teaching and/or don't particularly want to do it, taking lectures and tutorials.
I wanted to learn more about computer graphics, so I'm writing a 3D software renderer in C. So far I have a solid implementation of triangle rasterization, perspective projection, depth buffer, clipping, texture mapping, diffuse lighting, and gamma correction. Currently struggling with shadow mapping, which is the last feature I'll add to the renderer before moving on to procedural generation of meshes and textures.
Once I'm done with this project I'm planning on making a series of YouTube videos going into the code and the algorithms.
I got this itch too when I came across tinyrenderer [1] and worked through the early lessons through shading, but didn't quite finish the texture mapping yet [2]. It was fun to work in pure C from first principles, even side-questing to write a simple TGA file reader and writer.
I'd be very interested to see your tutorial when it's done!
You're assuming that the point of interactive theorem provers is to discover new mathematics. While that's an interesting research area, it seems like the more practical application is verifying proofs one has already discovered through other means.